Is symmetry identity?

Wigner found unreasonable the "effectiveness of mathematics in the natural sciences". But if the mathematics we use to describe nature is simply a coded expression of our experience then its effectiveness is quite reasonable. Its effectiveness is built into its design. We consider group theory, the logic of symmetry. We examine the premise that symmetry is identity; that group theory encodes our experience of identification. To decide whether group theory describes the world in such an elemental way we catalogue the detailed correspondence between elements of the physical world and elements of the formalism. Providing an unequivocal match between concept and mathematical statement completes the case. It makes effectiveness appear reasonable. The case that symmetry is identity is a strong one but it is not complete. The further validation required suggests that unexpected entities might be describable by the irreducible representations of group theory

Spontaneous symmetry breaking and bifurcations from the Maclaurin and Jacobi ellipsoids

The equilibrium of a rotating self-gravitating fluid is governed by non-linear equations. The equilibrium solutions, parametrized in terms of the angular momentum squared, exhibit the phenomenon of bifurcation, accompanied by spontaneous symmetry breaking. Under very general assumptions, a set of selection rules can be derived, which drastically restrict the patterns of symmetry breaking that are allowed to appear. Bifurcations of this kind are similar to second-order phase transitions à la Landau. The method is illustrated by the simple example of an incompressible fluid in rigid rotation. However, the selection rules are more general; they apply also to models which approximate a rotating star more realistically.L'état d'équilibre d'un fluide tournant soumis à son interaction gravitationnelle est déterminé par des équations non linéaires. Les solutions d'équilibre, paramétrées par le carré du moment cinétique, présentent des bifurcations accompagnées de brisures de symétrie. D'hypothèses très générales on déduit des règles de sélection concernant les brisures de symétrie qui peuvent apparaître dans ce problème. Les bifurcations sont du même type que celles à la Landau qui apparaissent dans les transitions de phase du second ordre. La méthode est illustrée par l'exemple simple d'un fluide incompressible animé d'une rotation globale et une nouvelle famille infinie de bifurcations est trouvée. Cependant les règles de sélection sont plus générales ; elles s'appliquent aussi aux modèles qui représentent la rotation d'une étoile de façon plus réaliste